Standard Lattices of Compatibly Embedded Finite Fields
Pages 122 - 130
Abstract
Lattices of compatibly embedded finite fields are useful in computer algebra systems for managing many extensions of a finite field \F_p at once. They can also be used to represent the algebraic closure \bar\F_p, and to represent all finite fields in a standard manner.
The most well known constructions are Conway polynomials, and the Bosma--Cannon--Steel framework used in Magma. In this work, leveraging the theory of the Lenstra-Allombert isomorphism algorithm, we generalize both at the same time.
Compared to Conway polynomials, our construction defines a much larger set of field extensions from a small pre-computed table; however it is provably as inefficient as Conway polynomials if one wants to represent all field extensions, and thus yields no asymptotic improvement for representing \bar\F_p.
Compared to Bosma--Cannon--Steel lattices, it is considerably more efficient both in computation time and storage: all algorithms have at worst quadratic complexity, and storage is linear in the number of represented field extensions and their degrees.
Our implementation written in C/Flint/Julia/Nemo shows that our construction in indeed practical.
References
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Wieb Bosma, John Cannon, and Allan Steel. 1997. Lattices of compatibly embedded finite fields. Journal of Symbolic Computation 24, 3--4 (1997), 351--369.
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Index Terms
- Standard Lattices of Compatibly Embedded Finite Fields
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Published In
July 2019
418 pages
ISBN:9781450360845
DOI:10.1145/3326229
- General Chairs:
- James Davenport,
- Dongming Wang,
- Program Chair:
- Manuel Kauers,
- Publications Chair:
- Russell Bradford
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Published: 08 July 2019
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ISSAC '19
ISSAC '19: International Symposium on Symbolic and Algebraic Computation
July 15 - 18, 2019
Beijing, China
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